Warped product submanifolds of Lorentzian paracosymplectic manifolds

Selcen Yüksel Perktaş,Sadık Keleş,Erol Kılıç

doi:10.1007/s40065-012-0037-y

Selcen Yüksel Perktaş, Sadık Keleş + Show 1 more

Open Access

https://doi.org/10.1007/s40065-012-0037-y

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Abstract

In this paper we study the warped product submanifolds of a Lorentzian paracosymplectic manifold and obtain some nonexistence results. We show that a warped product semi-invariant submanifold in the form $${M=M_{\top}\times _{f}M_{\bot}}$$ of a Lorentzian paracosymplectic manifold such that the characteristic vector field is normal to M is a usual Riemannian product manifold where totally geodesic and totally umbilical submanifolds of warped product are invariant and anti-invariant, respectively. We prove that the distributions involved in the definition of a warped product semi-invariant submanifold are always integrable. A necessary and sufficient condition for a semi-invariant submanifold of a Lorentzian paracosymplectic manifold to be warped product semi-invariant submanifold is obtained. We also investigate the existence and nonexistence of warped product semi-slant and warped product anti-slant submanifolds in a Lorentzian paracosymplectic manifold.

Highlights

Warped product manifolds were introduced by Bishop and O’Neill [7] in 1969 as a generalization of Riemannian product manifolds
The theory of slant immersions in complex geometry was introduced by Chen [12,13] as a generalization of both holomorphic and totally real submanifolds
Warped product semi-slant submanifolds in locally Riemannian product manifolds and Kenmotsu manifolds were studied by Atçeken [2,3], respectively

Summary

Introduction

Warped product manifolds were introduced by Bishop and O’Neill [7] in 1969 as a generalization of Riemannian product manifolds. As a generalization of warped product CR-submanifolds warped product semi-slant submanifolds are very important in differential geometry. Since every structure on a manifold may not allow defining warped product semi-slant submanifolds, the existence and nonexistence of these submanifolds are basic problems to study. Warped product semi-slant submanifolds in locally Riemannian product manifolds and Kenmotsu manifolds were studied by Atçeken [2,3], respectively. In [4], the author studied the warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds. In this paper we study warped product submanifolds of a Lorentzian paracosymplectic manifold and obtain to some nonexistence results. 4, we study warped product semi-invariant submanifolds of a Lorentzian paracosymplectic manifold and give an example. The last section contains some nonexistence results for the proper warped product anti-slant submanifolds of a Lorentzian paracosymplectic manifold

Preliminaries

Warped and doubly warped submanifolds

Warped product semi-invariant submanifolds

Warped product semi-slant submanifolds

Warped product anti-slant submanifolds